![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > tmslemOLD | Structured version Visualization version GIF version |
Description: Obsolete version of tmslem 24345 as of 28-Oct-2024. Lemma for tmsbas 24347, tmsds 24348, and tmstopn 24349. (Contributed by Mario Carneiro, 2-Sep-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
tmsval.m | β’ π = {β¨(Baseβndx), πβ©, β¨(distβndx), π·β©} |
tmsval.k | β’ πΎ = (toMetSpβπ·) |
Ref | Expression |
---|---|
tmslemOLD | β’ (π· β (βMetβπ) β (π = (BaseβπΎ) β§ π· = (distβπΎ) β§ (MetOpenβπ·) = (TopOpenβπΎ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6922 | . . . 4 β’ (π· β (βMetβπ) β π β dom βMet) | |
2 | tmsval.m | . . . . 5 β’ π = {β¨(Baseβndx), πβ©, β¨(distβndx), π·β©} | |
3 | df-ds 17228 | . . . . 5 β’ dist = Slot ;12 | |
4 | 1nn 12227 | . . . . . 6 β’ 1 β β | |
5 | 2nn0 12493 | . . . . . 6 β’ 2 β β0 | |
6 | 1nn0 12492 | . . . . . 6 β’ 1 β β0 | |
7 | 1lt10 12820 | . . . . . 6 β’ 1 < ;10 | |
8 | 4, 5, 6, 7 | declti 12719 | . . . . 5 β’ 1 < ;12 |
9 | 2nn 12289 | . . . . . 6 β’ 2 β β | |
10 | 6, 9 | decnncl 12701 | . . . . 5 β’ ;12 β β |
11 | 2, 3, 8, 10 | 2strbas 17176 | . . . 4 β’ (π β dom βMet β π = (Baseβπ)) |
12 | 1, 11 | syl 17 | . . 3 β’ (π· β (βMetβπ) β π = (Baseβπ)) |
13 | xmetf 24190 | . . . . 5 β’ (π· β (βMetβπ) β π·:(π Γ π)βΆβ*) | |
14 | ffn 6711 | . . . . 5 β’ (π·:(π Γ π)βΆβ* β π· Fn (π Γ π)) | |
15 | fnresdm 6663 | . . . . 5 β’ (π· Fn (π Γ π) β (π· βΎ (π Γ π)) = π·) | |
16 | 13, 14, 15 | 3syl 18 | . . . 4 β’ (π· β (βMetβπ) β (π· βΎ (π Γ π)) = π·) |
17 | 2, 3, 8, 10 | 2strop 17177 | . . . . 5 β’ (π· β (βMetβπ) β π· = (distβπ)) |
18 | 17 | reseq1d 5974 | . . . 4 β’ (π· β (βMetβπ) β (π· βΎ (π Γ π)) = ((distβπ) βΎ (π Γ π))) |
19 | 16, 18 | eqtr3d 2768 | . . 3 β’ (π· β (βMetβπ) β π· = ((distβπ) βΎ (π Γ π))) |
20 | tmsval.k | . . . 4 β’ πΎ = (toMetSpβπ·) | |
21 | 2, 20 | tmsval 24344 | . . 3 β’ (π· β (βMetβπ) β πΎ = (π sSet β¨(TopSetβndx), (MetOpenβπ·)β©)) |
22 | 12, 19, 21 | setsmsbas 24336 | . 2 β’ (π· β (βMetβπ) β π = (BaseβπΎ)) |
23 | 12, 19, 21 | setsmsds 24338 | . . 3 β’ (π· β (βMetβπ) β (distβπ) = (distβπΎ)) |
24 | 17, 23 | eqtrd 2766 | . 2 β’ (π· β (βMetβπ) β π· = (distβπΎ)) |
25 | prex 5425 | . . . . 5 β’ {β¨(Baseβndx), πβ©, β¨(distβndx), π·β©} β V | |
26 | 2, 25 | eqeltri 2823 | . . . 4 β’ π β V |
27 | 26 | a1i 11 | . . 3 β’ (π· β (βMetβπ) β π β V) |
28 | 12, 19, 21, 27 | setsmstopn 24341 | . 2 β’ (π· β (βMetβπ) β (MetOpenβπ·) = (TopOpenβπΎ)) |
29 | 22, 24, 28 | 3jca 1125 | 1 β’ (π· β (βMetβπ) β (π = (BaseβπΎ) β§ π· = (distβπΎ) β§ (MetOpenβπ·) = (TopOpenβπΎ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1084 = wceq 1533 β wcel 2098 Vcvv 3468 {cpr 4625 β¨cop 4629 Γ cxp 5667 dom cdm 5669 βΎ cres 5671 Fn wfn 6532 βΆwf 6533 βcfv 6537 1c1 11113 β*cxr 11251 2c2 12271 ;cdc 12681 ndxcnx 17135 Basecbs 17153 distcds 17215 TopOpenctopn 17376 βMetcxmet 21225 MetOpencmopn 21230 toMetSpctms 24180 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-fz 13491 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-tset 17225 df-ds 17228 df-rest 17377 df-topn 17378 df-topgen 17398 df-psmet 21232 df-xmet 21233 df-bl 21235 df-mopn 21236 df-top 22751 df-topon 22768 df-bases 22804 df-tms 24183 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |